What is an antiderivative and can 1 function have more than one antiderivatives?I want to know what the definition of an antiderivative is really. I know that the antiderivative of a function between 2 limits will give you the area under the graph between these limits. Take the function y = 1 / ( − 2 x ). The integral will be − ( 1 / 2 ) ln ⁡ | x | or it can also be − ( 1 / 2 ) ln ⁡ ( 2 | x | ) which is equal to − 0.5 ln ⁡ ( 2 ) − 0.5 ln ⁡ | x | . So as you can see they differ only by a constant. So when you plug in two limits you will get the same value. But i am wondering what an antiderivative represents exactly and how can it be possible as you saw to have a function with 2 antiderivatives. And by the way, can an antiderivative be represented by a graphical method?

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What is an antiderivative and can 1 function have more than one antiderivatives?I want to know what the definition of an antiderivative is really. I know that the antiderivative of a function between 2 limits will give you the area under the graph between these limits. Take the function   y  =  1      /    (  −  2  x  ). The integral will be   −  (  1      /    2  )  ln  ⁡      |    x      |   or it can also be   −  (  1      /    2  )  ln  ⁡  (  2      |    x      |    ) which is equal to   −  0.5  ln  ⁡  (  2  )  −  0.5  ln  ⁡      |    x      |  . So as you can see they differ only by a constant. So when you plug in two limits you will get the same value. But i am wondering what an antiderivative represents exactly and how can it be possible as you saw to have a function with 2 antiderivatives. And by the way, can an antiderivative be represented by a graphical method?

lisicw2 · Accepted Answer

Step 1If you understand a derivative, you will understand that the derivative of a constant is 0.You will also know that the derivative of   f  (  x  )  +  g  (  x  ) equals the derivative of f(x) plus the derivative of g(x), for appropriate conditions on f and g.That means that if c is a constant, the derivative of   f  (  x  )  +  c is the same as the derivative of f(x).So, suppose F(x) is the derivative of f(x).That is, f(x) is the antiderivative of F(x).Then F(x) is also the derivative of   f  (  x  )  +  c, whatever c is.And in this context, c can be any real number.That means there are as many different antiderivatives of F(x) as there are real numbers -- an uncountable infinity of them.It can also be shown (but it needs a bit more advanced thinking) that if f(x) and g(x) are both antiderivatives of F(x), then g(x) is always going to be   f  (  x  )  +  c, where c is a constant.That is, of all the antiderivatives of a given function, they are all the same except for being different by a constant.Step 2As for a graphical method, the antiderivative of f is the function whose gradient is f. You can get an approximation to it by picking a point in the plane (any point), say at   x  =      x    0  , and calculate   f  (      x    0    ). Then you draw a short arrow with that gradient. The end of that arrow is at   x  =      x    1  . You calculate   f  (      x    1    ) and draw another short arrow with the gradient equal to   f  (      x    1    ), that is, the gradient at that point. And so on, drawing a line made up of short arrows. As the length of the arrow tends to 0, the line represents the curve that is an antiderivative of f.

Taniyah Estrada · Accepted Answer

Explanation:An antiderivative is just a function whose derivative is the function you&#039;re after. So in your example,   −  0.5  ln  ⁡      |    x      |   and   −  0.5  ln  ⁡      |    2  x      |   are both antiderivatives of             −      1              2      x      . A function actually has infinitely many antiderivatives, because the   +  C you get can be any real number. The reason there are so many is that taking a derivative eliminates any constant terms, so adding a constant to an antiderivative doesn&#039;t change the fact that taking its derivative gets you back the original function.

What is an antiderivative and can 1 function have more than one antiderivatives? I want to know wha

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